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United States Patent |
5,177,700
|
Gockler
|
January 5, 1993
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Non-recursive half-band filter
Abstract
A non-recursive half-band filter having a filter length N and complex
coefficients for receiving either a real input signal s(kT) or a complex
input signal s(2kT) and for processing and converting the received input
signals into either a complex output signal s(2kT) or a real output signal
s(kT) wherein the complex coefficients operate at a function of h(l) where
l=-(N-1)/2 to (N-1)/2 to (N-1)/2 and the filter length N is odd. As a
result it is possible to convert a real input signal into a complex output
signal, by modulating its pulse response to a complex carrier of the
frequency equal to 1/4 or 3/4 of the sampling frequency, where the null
phase of this frequency is an integer multiple of .pi./2. It is also
possible to convert a complex input signal into a real output signal, by
modulating its pulse frequency to the complex carrier of a frequency
signal to the input sampling frequency or half thereof, where the null
phase of this frequency is an integer multiple of .pi./2.
Inventors:
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Gockler; Heinz (Backnang, DE)
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Assignee:
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ANT Nachrichtentechnik GmbH (Backnang, DE)
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Appl. No.:
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814265 |
Filed:
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December 24, 1991 |
PCT Filed:
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December 24, 1987
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PCT NO:
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PCT/DE87/00607
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371 Date:
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August 18, 1989
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102(e) Date:
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August 18, 1989
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PCT PUB.NO.:
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WO88/06380 |
PCT PUB. Date:
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August 25, 1988 |
Foreign Application Priority Data
Current U.S. Class: |
708/313; 708/316; 708/319 |
Intern'l Class: |
G06F 015/31 |
Field of Search: |
364/724.1,724.13,724.16
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References Cited
U.S. Patent Documents
4344149 | Aug., 1982 | Meeberg et al. | 364/724.
|
4612625 | Sep., 1986 | Bertrand | 364/724.
|
4766562 | Aug., 1988 | Vary | 364/724.
|
4825396 | Apr., 1989 | Gazsi | 364/724.
|
4852035 | Jul., 1989 | Michener | 364/724.
|
4893264 | Jan., 1990 | Noll et al. | 364/724.
|
4896320 | Jan., 1990 | Gockler | 364/724.
|
Other References
Bellanger et al, "Interpolation, Extrapolation, and Reduction of
Computation Speed in Digital Filters" IEEE Trans. on Acoustics, Speech &
Signal Processing vol. ASSP-22, No. 4 Aug. 1974 pp. 231-235.
Ansari, "Elliptic Filter Design for a Class of Generalized Halfband
Filters", IEEE Trans. on Acoustics, Speech & Signal Processing vol.
ASSP-33 No. 4 Oct. 1985 pp. 1146-1150.
|
Primary Examiner: Malzahn; David H.
Attorney, Agent or Firm: Spencer, Frank & Schneider
Parent Case Text
This application is a continuation of application Ser. No. 07/408,493,
filed Aug. 18, 1989, now abandoned.
Claims
I claim:
1. A non-recursive half-band filter having a filter length N for processing
a real input signal s(kT) and converting the real input signal into a
complex output signal s(2kT), said half-band filter comprising:
input means for receiving samples of a real input signal s(kT) at a
sampling frequency fA, where fA=1/T, and for providing samples at one-half
of the sampling frequency fA; and
means, responsive to the provided samples, for processing and converting
the real input signal into a complex output signal s(2kT), said means
having complex coefficients h(l), where l=-(N-1)/2 to (N-1)/2 and the
filter length N is odd, with alternating purely real and purely imaginary
values, said processing and converting means including means for
modulating the pulse response of a half-band filter h(l) with exclusively
real values and h(l) =h(-l) for all .vertline.l.vertline..ltoreq.(N-1)/2
and h(l)=0 for l=.+-.2, .+-.4, . . . , onto a complex carrier of a
frequency of .+-.1/4 of the input sampling frequency to yield
h(l)=h(l).multidot.e.sup.j(.+-.2.pi.lfA/4fA+.phi.0) =j.sup..+-.l
.multidot.e.sup.j.phi.0 .multidot.h(l),
wherein the null phase .phi.0 of the complex carrier is an integer multiple
m of .pi./2 (.phi.0=m.multidot..pi./2 where m=0, 1, 2, 3 . . .), and
wherein said modulating means is divided into a first branch and a second
branch, said first branch comprising a chain of (N-1)/2 delay members each
have a delay time of 2T, means for forming a plurality of difference
signals and means for weighting said difference signals, with every second
sample of the input signal s(kT) being routed into the chain of (N-1)/2
delay members; said difference signal forming means subtracting from the
output signal of the last delay member in the chain the input signal of
the first delay member of the chain to form a first difference signal,
subtracting from the output signal of the penultimate delay member in the
chain the input signal of the second delay member of the chain to form a
second difference signal, subtracting from the output signal of the third
to last delay member in the chain the input signal of the third delay
member of the chain to form a third difference signal, and so on until the
outputs of each delay member are processed; said weighting means including
an adder and weighting the respective said difference signals by a
function of h(l) of the pulse response and summing the weighted signals
via the adder to yield either the real or the imaginary component of the
filter output signal s(2kT); said second branch including a second branch
delay member which has a time delay of T.multidot.(N-3)/2, into which is
routed every other sample of the input signal, the output signal of said
second branch delay member being weighted with a value equal to a function
of h(0) to yield the other of the imaginary component and the real
component of the filter output signal s(2kT).
2. A non-recursive half-band filter according to claim 1, where N=11 and
m=1, and said weighting means weights the first difference signal with a
value equal to the function -h(5), the second difference signal with a
value equal to the function h(3) and the third difference signal a value
equal to the function -h(1); and the sum of the difference signals yields
the real component s.sub.r (2kT) of the filter output signal while the
signal weighted with the function h(0)=1/2 yields the imaginary component
s.sub.i (2kT) of the filter output signal.
3. A non-reactive half-band filter according to claim 1, where N=11 and
m=3, and said weighting means weights the first difference signal with a
value equal to the function h(5), the second difference signal with a
value equal to the function -h(3), and the third difference signal with a
value equal to the function h(1); and the sum of the difference signals
yields the real component s.sub.r (2kT) while the signal weighted with the
function h(0)=-1/2 yields the imaginary component s.sub.i (2kT) of the
filter output signal.
4. A non-recursive half-band filter according to claim 1, wherein N=11 and
m=0, and said weighting means weights the first difference signal with a
value equal to the function h(5), the second difference with a value equal
to the function -h(3), and the third difference signal with a value equal
to the function h(l); and the sum of the difference signals yields the
imaginary component s.sub.i (2kT) of the filter output signal while the
signal weighted with the function h(0)=1/2 yields the real component
s.sub.r (2kT) of the filter output signal.
5. A non-recursive half-band filter according to claim 1, where N=11 m=2,
and said weighting means weights the first difference signal with a value
equal to the function -h(5), the second difference signal with a value
equal to the function h(3), and the third difference signal with a value
equal to the function -h(l); and the sum of the difference signals yields
the imaginary component s.sub.i (2kT) while the signal weighted with the
function h(0)=1/2 yields the real component s.sub.r (2kT) of the filter
output signal.
6. A non-recursive half-band filter having a filter length N for processing
a complex input signal s(2kT) and for converting the complex input signal
for into a real output signal s(kT) at double the input sampling
frequency, said half-band filter comprising:
input means for receiving samples of a complex input signal s(2kT) at a
sampling frequency fA'=1/2T; means for processing and converting the
received complex input signal into a real output signal s(kT) at an output
sampling frequency fA=2fA' said means having complex coefficients h(l),
where l=-(N-1)/2 to (N-1)/2 and the filter length N is odd, with
alternating purely real and purely imaginary values, said processing and
converting means including means for modulating the pulse response of a
half-band filter h(l) with exclusively real values and h(l)=h(-l) for all
.vertline.l.vertline..ltoreq.(N-1)/2 and h(l)=0 for l=.+-.2, .+-.4, . . .
, onto a complex carrier of a frequency of .+-.1/4 of the output sampling
frequency fA=2fA' to yield
h(l)=h(l).multidot.e.sup.j(.+-.2.pi.lfA/4fA+.phi.0) =j.sup..+-.l
.multidot.e.sup.j.phi.0 .multidot.h(l),
with the null phase .phi.0 of the complex carrier being an integer multiple
m of .pi./2 (.phi.0=m.multidot..pi./2 where m=0, 1, 2, 3, . . .); and
wherein said complex input signal includes a real component and an
imaginary component, and said processing and converting means is divided
into first and second branches, said first branch comprising a chain
(N-1)/2 delay members each having a time delay of 2T, means for weighting
said real and imaginary components of the complex input signal and means
for forming a plurality of difference signals, the weighting means weights
the real component s.sub.r (2kT) with a value equal to a function of h(l)
of the pulses response and said forming means feeds the weighted component
to the first delay member of the chain of delay members and subtracts the
weighted component from the output signal of the last delay member of the
chain to produce a difference signal, said difference signal furnishing
every second sample of the real filter output signal s(kT); said weighting
means weights additional momentary values of the real component s.sub.r
(2kT) of the filter input signal with a value equal to other functions of
h(l) of the pulse response and said forming means adds these additional
weighted values to a transversal signal of the chain of delay members at
further points; and said second branch has a further delay member with a
time delay of T.multidot.(N-3)/2 whose input receives the imaginary
component s.sub.i (2kT) of the complex input signal which has been
weighted with a value equal to a function of h(0) and whose output yields
every second time-shifted sample of the real filter output signal s(kT).
7. A non-reactive half-band filter according to claim 6, where m=0 and
N=11, and the weighting means weights the additional momentary values of
the real component s.sub.r (2kT) of the filter input signal and the
forming means adds the weighted values to the transversal signal to the
points of the chain as follows:
at the input of the first delay member of the chain, the real component is
weighted with a value equal to the function h(5);
at the input of the second delay member of the chain, the real component is
weighted with a value equal to the function -h(3);
at the input of the third delay member of the chain, the real component to
weighted with a value equal to the function h(1);
at the input of the fifth delay member of the chain, the real component is
weighted with a value equal to the function h(3); and
at the output of the fifth delay member of the chain, the real component is
weighted with a value equal to the function -h(5); where h(0)=1/2.
8. A non-recursive half-band filter having a pulse response h(l), where
l=-(N-1)/2 to (N-1)/2 and N is an odd filter length, said filter
comprising means for receiving a real input signal s(kT) and means for
converting the real input signal s(kT) into a complex output signal s(kT),
where k is a running index, while maintaining a sampling frequency fA=1T,
said converting means modulating the pulse response h(l) onto a complex
carrier with a frequency of .+-.1/4 of the sampling frequency fA=1/T, to
yield
h(l)=h(l).multidot.e.sup.j(.+-.2.pi.lfA/4fA'.phi.0) =j.sup..+-.l
.multidot.e.sup.j.phi.0 .multidot.h(l),
with the null phase .phi. 0 of this frequency being an integer multiple m
of .pi./2 (.phi.0=m.multidot..pi./2 where m=1, 2, 3, . . .), and wherein
said converting means includes a chain of (N-1)/2 delay members having a
center delay member, means for forming a plurality of difference signals
and means for weighting the formed difference signals, where each sample
of the input signal s(kT) is routed into the chain of (N-1)/2 delay
members each having a delay time of 2T, and the center delay member is
divided into two members each having a delay time of T; said difference
signal forming means subtracting from the output signal of the last delay
member of the chain the input signal of the first delay member of the
chain to form a first difference signal, subtracting from the output
signal of the penultimate delay member of the chain the input signal of
the second delay member of the chain to form a second difference signal,
subtracting from the output signal of the third to last delay member of
the chain the input signal of the third delay member of the chain to form
a third difference signal, and so on until the output signal of each delay
member is processed; said weighting means includes an adder and weighting
each said difference signal by a value equal to a respective function of
h(l) of the pulse response and summing the weighted difference signals via
the adder to yield either the real or the imaginary component of the
filter output signal s(kT); said forming means forms a fourth difference
signal from the center delay member of the chain where the input signal is
delayed by a delay time T.multidot.(N-1)/2 and is weighted with a value
equal to the function j(0), which results in the other of the imaginary
component and the real component of the filter output signal s(kT).
9. A non-reactive half-band filter according to claim 8, where N=11 and
m=1, and the weighted means weights the first difference signal with a
value signal equal to the function -h(5), the second difference signal
with a value equal to the function h(3) and third difference signal with a
value equal to the function -h(1); and the sum of the difference signals
yields the real component s.sub.r (kT) of the filter output signal while
the signal weighted with the function of h(0)=1/2 yields the imaginary
component s.sub.i (kT) of the filter output signal.
10. A non-recursive half-band filter according to claim 8, where N=11 and
m=3, and the weighting means weights the first difference signal with a
value equal to the function h(5), the second difference signal with a
value equal to the function -h(3), the third difference signal with a
value equal to the function h(1); and the sum of the difference signals
yields the real component s.sub.r (kT) of the filter output signal while
the signal weighted with the function of h(0)=1/2 yields the imaginary
component s.sub.i (kT) of the filter output signal.
11. A non-reactive half-band filter according to claim 8, wherein N=11 m=0,
and the weighting means weights the first difference signal with a value
equal to the function h(5), the second difference signal with a value
equal to the function -h(3), and the third difference signal with a value
equal to the function h(1); and the sum of the difference signals yields
the imaginary component s.sub.i (kT) of the filter output signal while the
signal weighted with the function of h(0)=1/2 yields the real component
s.sub.r (kT) of the filter output signal.
12. A non-recursive half-band filter according to claim 8, where N=11 and
m=2, and the weighting means weights the first difference signal with a
value equal to the function -h(5), the second difference signal with a
value equal to the function h(3), the third difference signal with a value
equal to the function -h(1); and the sum of the difference signals yields
the imaginary component s.sub.i (kT) of the filter output signal while the
signal weighted with the function h(0)=-1/2 yields the real component
s.sub.r (kT) of the filter output signal.
13. A non-recursive half-band filter having a filter length N and a pulse
response h(l) where l=-(N-1)/2 to (N-1)/2 and the filter length N is odd,
said half-band filter comprising: means for receiving a complex input
signal, and means for converting said complex input signal s(kT), where k
is a running index, into a real output signal s(kT), while maintaining a
sampling frequency fA=1/T, said converting means modulating the pulse
response h(l) with reference to the sampling frequency fA, onto a complex
carrier at a frequency of .+-.fA/4 to yield
h(l)=h(l).multidot.e.sup.j(.+-.2.pi.lfA/4fA+.phi.0) =j.sup..+-.l
.multidot.e.sup.j.phi.0 .multidot.h(l)
with the null phase .phi. 0 of this frequency being an integer multiples m
of .pi./2 (.phi.0=m.multidot..pi./2 where m=0, 1, 2, 3, . . .), and
wherein said complex input signal has a real component and an imaginary
component; and said converting means includes a chain of (N-1)/2 delay
members each having a time delay of 2T and a center delay member divided
into two members, with each divided member having a delay time of T, means
for weighting the real and imaginary components with a value equal to a
function of h(l), and means for forming a plurality of difference signals;
and the weighting means weights the imaginary component s.sub.i (kT) with
values equal to functions of h(l) of the pulse response and said forming
means feeds the weighted component to the first delay member of the chain
and subtracts the weighted component from the output signal of the last
delay member of the chain to form a difference signal, said difference
signal furnishing the real filter output signal s(kT); said weighting
means weights additional momentary values of the imaginary component
s.sub.i (kT) of the filter input signal with a value equal to other
functions of h(l) of the pulse response and said forming means adds the
additional weighted values to a transversal signal of the chain of delay
members at further points; and the weighting means further weights the
real component s.sub.r (kT) of the complex filter input signal with the
function h(0), and additively feeds the weighted signal to the transversal
signal of the chain at the center of said center delay member.
14. Non-recursive half-band filter according to claim 13, where m=0 and 2,
respectively, and N=11, and the weighting means weights the additional
momentary values of the real component s.sub.r (kT) and of the imaginary
component s.sub.i (kT), respectively, of the filter input signal and the
forming means adds the weighted values to the transversal signal to the
points of the chain as follows:
at the input of the first delay member of the chain, with a value equal to
the function .+-.h(5);
at the input of the second delay member of the chain with a value equal to
the function .+-.h(3);
at the input of the third delay member of the chain with a value equal to
the function .+-.h(1);
at the input of the penultimate delay member of the chain with a value
equal to the function .+-.h(1);
at the input of the last delay member of the chain with a value equal to
the function .+-.h(3); and
at the output of the last delay member of the chain with a value equal to
the function .+-.h(5); where h(0)=.+-.1/2.
Description
BACKGROUND OF THE INVENTION
The invention relates to a non-recursive half-band filter. Such filters
have become known from the paper by Bellanger et al, entitled,
"Interpolation, Extrapolation, and Reduction of Computation Speed in
Digital Filters," published in IEEE Transactions on Acoustics, Speech and
Signal Processing, Vol. ASSP-22, No. 4, August, 1974, pages 231-235.
The known half-band filters process real input signals into real output
signals.
SUMMARY OF THE INVENTION
It is an object of the present invention to provide a non-recursive
half-band filter that makes it possible to convert a real input signal
into a complex output signal, or vice versa, in an inexpensive manner.
The above object is achieved according to a first aspect of the invention
by a non-recursive half-band filter with complex coefficients for
processing a real input signal s(kT) by having the sampling frequency
fA=1/T and for converting this real input signal s(kT) into a complex
output signal s(2kT), wherein the filter complex coefficients h(l), where
l=-(N-1)/2 to (N-1)/2 and the filter length N is odd, have alternating
purely real and purely imaginary values, and therefore no complex values
in the fullest sense, and wherein the pulse response of a half-band filter
h(l) having exclusively real values and the characteristics h(l)=h(-l) for
all .vertline.l.vertline..ltoreq.(N-1)/2 and h(l)=0 for 1=.+-.2, .+-.4, .
. . , is modulated onto the complex carrier of a frequency of .+-.1/4 of
the input sampling frequency fA=1/T to yield
h(l)=h(l).multidot.e.sup.j(.+-.2.pi.lfA/4fA+.phi.0) =j.sup..+-.l
.multidot.e.sup.j.phi.0 .multidot.h(l)
and the null phase .phi.0 of this complex carrier is an integer multiple m
of .pi./2 (.phi.0=m.multidot..pi.2 where m=0, 1, 2, 3, . . . ).
The above object is achieved according to another aspect of the invention
by a non-recursive half-band filter with complex coefficients for
processing a complex input signal s(2kT) and for doubling the sampling
frequency fA'=1/2T to fA=2fA' and for converting this complex input signal
s(2kT) into a real output signal s(kT), and wherein the filter complex
coefficients h(l), where l=-(N-1)/2 to (N-1)/2 and the filter length N is
odd, alternatingly have purely real and purely imaginary values, and
therefore no complex values in the fullest sense, and the pulse response
of a half-band filter h(l) having exclusively real values and the
characteristics h(l)=h(l) for all .vertline.l.vertline..ltoreq.(N-1)/2 and
h(l)=0 for l=.+-.2, .+-.4, . . . , is modulated onto the complex carrier
of a frequency of .+-.1/4 of the output sampling frequency fA=1/T to yield
h(l)=h(l).multidot.e.sup.j(.+-..pi.lfA/4fA)+.phi.0) =j.sup..+-.l
.multidot.e.sup.j.phi.0 .multidot.h(l)
and the null phase .phi.0 of this complex carrier is an integer multiple m
of .pi./2 (.phi.0=m.multidot..pi./2 where m=0, 1, 2, 3, . . . ).
The above object is achieved according to still a further object of the
invention by a non-recursive half-band filter, which converts a real input
signal s(kT) into a complex output signal s(kT) where k is a running
index, while maintaining the sampling frequency fA=1/T, in that the filter
pulse response h(l), where l=-(N-1)/2 to (N-1)/2 and N is an odd filter
length, is modulated onto a complex carrier at a frequency of .+-.1/4 of
the sampling frequency fA=1/T, to yield
h(l)=h(l).multidot.e.sup.j(.+-.2.pi.lfA/4fA+.phi.0) =j.sup..+-.l
.multidot.e.sup.j.phi.0 .multidot.h(l)
and the null phase .phi.0 of this frequency is an integer multiple m of
.pi./2 (.phi.0=m.multidot..pi./2 where m=0, 1, 2, 3, . . . ).
The above object is achieved according to still a further aspect of the
invention by a non-recursive half-band filter, which converts a complex
input signal s(kT) where k is a running index into a real output signal
s(kT), while maintaining the sampling frequency fA=1/T, in that the filter
pulse response h(l) with reference to the sampling frequency fA, where
l=-(N-1)/2 to (N-1)/2 and the filter length N is odd, is modulated onto
the complex carrier at a frequency of .+-.fA/4 to yield
h(l)=h(l).multidot.e.sup.j(.+-.2.pi.lfA/4fA+.phi.0) =j.sup..+-.l
.multidot.e.sup.j.phi.0 .multidot.h(l)
and the null phase 0 of this frequency is an integer multiple m of .pi./2
(.phi.0=m.multidot..pi./2 where m=0, 1, 2, 3, . . . ).
The novel non-recursive half-band filter according to each of the first two
aspects of the invention permits the conversion of real digital input
signals into complex digital output signals with a simultaneous reduction
of the sampling frequency by a factor of two, and the conversion of
complex digital input signal into real digital output signals with a
simultaneous increase in the sampling frequency by a factor of 2. The
novel non-recursive half-band filter according to the latter two aspects
of the invention permits the conversion of real digital input signals into
complex digital output signals while maintaining the sampling frequency,
and the conversion of complex digital input signals into real digital
output signals, likewise while maintaining the sampling frequency.
These relatively inexpensive half-band filters are thus suitable as digital
pre-filters or post-filters for digital systems employed to process
complex signals and as digital partial filters in an arrangement of
anti-aliasing filters for band limitation while complying with the
sampling theorem. The advantage of the half-band filter lies in its linear
phase and simultaneously its low cost. In each case, the smallest possible
sampling frequency required on the basis of the sampling theorem can be
employed.
BRIEF DESCRIPTION OF THE DRAWINGS
There now follows a description with reference to the drawing figures in
which:
FIG. 1 is a block circuit diagram for the digital filter according to the
invention.
FIGS. 2a to 2c depict several amplitude responses of half-band filters
plotted over frequency.
FIGS. 3 and 4 show particularly favorable circuit variations of the
half-band filter.
FIG. 5 is a block circuit diagram for a half-band filter used to process a
complex input signal into a real output signal, according to the present
invention.
FIG. 6 shows, in schematic, circuit details of the filter shown in FIG. 5,
with this circuit having been developed by transposition from that of FIG.
3, i.e., by reversing all directions indicated by arrows and replacing a
branching switch for an adder and vice versa, and by replacing a
demultiplexer with a multiplexer.
FIG. 7 shows, in schematic, another circuit arrangement of the filter shown
in FIG. 5, this circuit arrangement being developed from that of FIG. 4 in
a manner similar to that of FIG. 8.
FIG. 8 is another embodiment of a block circuit diagram for the digital
filter according to the invention.
FIGS. 9a to 9c show several amplitude responses of half-band filters
plotted over frequency.
FIGS. 10 and 11 show particularly favorable circuit variations of the
half-band filter.
FIG. 12 is a block circuit diagram of a transposed, inversely operated
half-band filter for processing a complex input signal into a real output
signal.
FIGS. 13 and 14 respectively show, in schematic, circuit details of the
filter according to FIG. 12, with the circuits being respectively
developed by transposition from FIGS. 10 and 11, i.e. by reversal of all
directions indicated by arrows and the replacement of a branching switch
with an adder and vice versa.
DESCRIPTION OF THE PREFERRED EMBODIMENT
In FIG. 1, the real input signal s(kT) is applied by halving the sampling
rate to a digital half-band filter DF which generates therefrom the
complex output signal s(2kT).
The amplitude frequency response of a prototype half-band filter is shown
in FIG. 2a; the pass band of this filter extends from -fA/4+.DELTA.f to
+fA/4-.DELTA.f, and its stop band also has a width of fA/2-2.DELTA.f. It
is a further characteristic of this half-band filter that the transition
from the stop band to the pass band is steady and takes place over a width
of 2.DELTA.f. This transition range is symmetrical to fA/4. A further
characteristic of the half-band filter is that its ripple in the pass band
and in the stop band is identical, namely .delta.1=.delta.2=.delta.. In
such a filter, there results a pulse response h(l) where 1=0 to N-1 and
the filter length N is odd, and it follows that every second value is
identical to zero, with the exception of the central main value (see FIG.
2, page 233, in the abovecited, paper by Bellanger et al).
FIG. 2b shows the frequency response .vertline.H.vertline.. It can be seen
that this frequency response has been shifted to the right by the
frequency fA/4 relative to the frequency response of the prototype
half-band filter. In addition, FIG. 2b shows that the spectrum
.vertline.S.vertline. of a real input signal s(kT) sampled at the sampling
frequency fA has been inserted; because of sampling with fA, this input
signal spectrum is periodically repeated in frequency ranges
[m.multidot.fA, (m+1/4).multidot.fA] in the normal position and in
frequency ranges [(m+1/4).multidot.fA, (m+1).multidot.fA] in the inverted
position where m=. . . , -1, 0, +1, . . . The input signal s(kT), applied
to the half-band filter according to the invention without any change in
the sampling rate would thus suppress the inverted position between fA/2
and fA and of course all its repetitions and would simultaneously generate
a complex signal s(kT). Halving the sampling rate now results in the
desired spectra, with the normal position being repeated in each instance
in a pattern of fA/2=fA', where fA' is the new sampling rate (see FIG.
2c).
At this point, it should be noted that a complex signal in the inverse
position is obtained at the output of the half-band filter if the
frequency response of the prototype half-band filter according to FIG. 2a
is shifted by -fA/4 or, equivalently, by +3fA/4.
FIG. 3 now shows a detailed embodiment of a half-band filter according to
the invention.
First, however, it should be noted, with reference to FIGS. 2a-c that the
halving of the sampling rate is carried out only after filtering. This
sequence for the procedure according to FIGS. 2a-c should be formally
adhered to. However, according to the invention, the half-band filter can
be divided into two branches, each of which is supplied from the start
with every second sample of the input signal. However, this means nothing
other than that the halving of the sampling rate can take place directly
at the filter input, as shown schematically in the block circuit diagram
of FIG. 1.
Accordingly, the detailed circuit embodiments of FIGS. 3 and 4 include an
input-side demultiplexer switch Sw which supplies the input signal s(kT)
to the upper branch and then to the lower branch, in each case at the
rhythm of the sampling rate fA'=fA/2.
Both FIG. 3 and FIG. 4 show, as an example, a realization for a filter
length of N=11. Accordingly, the lower branch incorporates a delay member
4T with a time delay of (N-3).multidot.T/2=4T, while the upper branch
includes a chain of five delay members 2T with a time delay of 2T.
The circuit arrangement of FIG. 3 can be employed for two variations namely
for a modulation phase angle .phi.0=0 and .phi.0=.pi. corresponding to m=0
and m=2. The output signal of the delay member of the lower branch is
weighted (multiplied) with h(0)=1/4 and thus yields the real component
S.sub.r (2kT) of the output signal. For m=2 weighting occurs with -1/4.
The further processing of the upper branch now takes place in such a way
that (N+1)/4=3 difference signals are formed:
the first difference signal equals the output signal of the first delay
member minus the input signal of the last delay member;
the second difference signal equals the output signal of the second delay
member minus the input signal of the penultimate delay member; and
the third difference signal equals the output signal of the third delay
member minus the input signal of the third last, i.e. the middle, delay
member.
Next, these difference signals are weighted (multiplied) and summed by
adder A and thereby yield the imaginary component of output signal s(2kT).
The weighting is effected according to the following tables.
Examples for N=11 and h(-l)=h(l), where l=0, 1, . . . , 5, corresponding to
the prototype half-band filter according to the frequency response curve
of FIG. 2a:
TABLE 1
______________________________________
m = 0 (for m = 2 in each case with the opposite sign for the
complex coefficients h = Re(h) + jJm(h))
l -5 -3 -1 0 1 3 5
______________________________________
Re(h) 0 0 0 h(0) 0 0 0
Im(h) -h(5) h(3) -h(1) 0 h(1) -h(3) h(5)
______________________________________
TABLE 2
______________________________________
m = 1 (for m = 3 in each case with the opposite sign for the
complex coefficients)
l -5 -3 -1 0 1 3 5
______________________________________
Re(h) h(5) -h(3) h(1) 0 -h(1) h(3) -h(5)
Im(h) 0 0 0 h(0) 0 0 0
______________________________________
The realization according to FIG. 4 takes place in the same manner as that
in FIG. 3; the sole difference is in the other null phase value
.phi.0=m.multidot..pi./2 where m=1 and 3, the only consequence of which is
a different weighting and an exchange of filter branch outputs.
FIG. 5 shows the block circuit diagram for the reversed use of the
half-band filter of FIG. 1, namely for the generation of a real output
signal from a complex input signal. To this end, there must be a
transposition of the circuits presented above, which results in a reversal
of the directions of all arrows and the replacement of a branching switch
BS.sub.w for adder A and vice versa, as well as the replacement of a
demultiplexer with a multiplexer. In a corresponding manner, the circuit
embodiment of FIG. 6 is derived from FIG. 3 and the circuit of FIG. 7 is
derived from FIG. 4. Thus, both FIGS. 6 and 7 show, as an example, a
realization for a filter length N=11 where m=0 or 2 in FIG. 6, and m=1 or
3 in FIG. 7.
In FIG. 8, the real input signal s(kT) is fed to digital half-band filter
DF which generates therefrom the complex output signal s(kT).
FIG. 9a shows the amplitude frequency response of a prototype half-band
filter; its pass band extends from -fA/4 +.DELTA.f to +fA/4-.DELTA.f (half
value) and its stop band also has a width of fA/2 -2.DELTA.f. It is a
further characteristic of the half-band filter that the transition from
the stop band to the pass band is steady and takes place over a width of
2.DELTA.f. This transition region is symmetrical to fA/4. A further
characteristic of the half-band filter is that its ripple is the same in
the pass band as in the stop band, namely .delta.1=.delta.2 =.delta.. In
such a filter, there results a pulse response h(l) where l=0 to N-1 and
the filter length N is odd with the result that every second value is
equal to zero, except for the central main value (see in this connection
also FIG. 2 at page 233 of the above-cited paper by Bellanger et al).
FIG. 9b shows the frequency response .vertline.H.vertline.. It can be seen
that this frequency response is shifted to the right by the frequency fA/4
relative to the frequency response of the prototype half-band filter. FIG.
9b additionally shows the spectrum .vertline.S.vertline. of a real input
signal s(kT) sampled at sampling frequency fA. Due to the sampling at fA,
this signal is periodically repeated in frequency ranges [m.multidot.fA,
(m+1/2).multidot.fA] in the normal position and in frequency ranges
[(m+1/2).multidot.fA, (m+1).multidot.fA] in the inverse position where m=.
. . , -1, 0 +1, . . . Thus the inverse position of real input signal
s(kT), applied to the half-band filter according to the invention without
a change in sampling rate, and of course all of its repetitions are
suppressed between fA/2 and fA and at the same time a complex signal s(kT)
is generated, (see FIG. 9c).
At this point it should be mentioned that a complex signal in the inverse
position is obtained at the output of the half-band filter if the
frequency response of the prototype half-band filter of FIG. 9a is shifted
by -fA/4 or, the equivalent, by +3fA/4.
FIG. 10 now shows a detailed embodiment of a half-band filter according to
the invention.
FIG. 10 as well as FIG. 14 shows exemplary realizations for a filter length
N=11 including a chain of six delay members (2T, T), four of which having
a delay time of 2T and two, which are disposed symmetrically between the
other four delay members, a delay time of T.
The circuit of FIG. 10 can be employed for two realizations of the
invention, namely for a modulation phase angle .phi.0=0 and .phi.0=.pi.,
corresponding to m =0 and m=2. The output signal of the delay members of
the left half of the chain is weighted (multiplied) with h(0)=1/2 and thus
provides the real component s.sub.r (kT) of the output signal. For m=2,
weighting occurs with -1/2. The further processing in the delay chain now
takes place in such a way that (N+1)/4=3 difference signals are formed:
the first difference signal is equal to the input
signal of the first delay member minus the output signal
of the last delay member;
the second difference signal is equal to the input signal of the second
delay member minus the output signal of the penultimate delay member; and
the third difference signal is equal to the input signal of the third delay
member minus the output signal of the third last, i.e. the middle, delay
member on the right.
Next, these difference signals are weighted (multiplied) and summed by an
adder A and thereby yield the imaginary component of output signal s(kT)
The weighting is effected according to the following tables.
Examples for N=11 and h(-l)=h(l), where l=0, 1, . . . 5, corresponding to
the prototype half-band filter according to the frequency response curve
of FIG. 2a:
TABLE 11
______________________________________
m = 0 (for m = 2 in each case with the opposite sign for the
complex coefficients h = Re(h) + jJm(h))
l -5 -3 -1 0 1 3 5
______________________________________
Re(h) 0 0 0 h(0) 0 0 0
Jm(h) -h(5) h(3) -h(1) 0 h(1) -h(3) h(5)
______________________________________
TABLE 12
______________________________________
m = 1 (for m = 3 in each case with the opposite sign for the
complex coefficients)
l -5 -3 -1 0 1 3 5
______________________________________
Re(h) h(5) -h(3) h(1) 0 -h(1) h(3) -h(5)
Jm(h) 0 0 0 h(0) 0 0 0
______________________________________
The realization according to FIG. 11 takes place in the same manner as that
in FIG. 10; the sole difference is in the other null phase value
.phi.0=m.multidot..pi./2 where m=1 and 3, the only consequence of which is
a different weighting.
FIG. 12 shows the block circuit diagram for the reversed use of the
half-band filter of FIG. 8, namely for the generation of a real output
signal from a complex input signal. For this purpose, there must be a
transposition of the circuits presented above, which results in a reversal
of the directions of all arrows and the replacement of a branching switch
BS.sub.w adder A and vice versa. In a corresponding manner, the circuit
embodiment of FIG. 13 is derived from FIG. 10 and the circuit of FIG. 14
is derived from FIG. 11. Accordingly, both FIGS. 13 and 14 show a
realization for a filter length N=11 where m=0 or 2 in FIG. 13, and m=1 or
3 in FIG. 14.
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